The notation in eq.(REF ) is, by design, suggestive: the FP equation, written this way, looks very much like one half of Hamilton's equations. And, while this will eventually be the case, it is important to guard against the conclusion that the FP equation was derived from an action principle. It was not. The evolution of \(\rho \)  has been derived, through and through, from a completely inferential framework; it is only once we consider the full joint dynamics of \(\rho \)  and \(\Phi \)  together that \(\tilde{H}[\rho ,\Phi ]\)  obtains its interpretation as a true Hamiltonian functional.The observant among us will have noticed that there was a bit of subjectivity in choosing the FP equation to be written as in (REF ) since it automatically assumes \(\rho (x)\)  to be the canonical coordinate rather than the conjugate momentum. The choice is not, however, unreasonable. For example, this follows naturally when the e-phase space is modeled as a cotangent bundle, as in [1]}. Alternatively, had we made the opposite choice, the e-functional \(\tilde{H}\)  would've been negative, and thus we could not have identified the generator \(\tilde{H}\)  as the “energy". A convenient convention is then to choose the generator of time translations to coincide with the energy, as the definition eq.(REF ) then guarantees.
